Okay, so good morning to the Tuesday session, the last session before Christmas.

Today we will continue to talk about CT reconstruction, computed tomography methods.

But before we dig into the next subsection on this topic, let's briefly summarize what

this semester is about.

Basically, we have four core topics.

One was just the introduction where we discussed modalities.

The second one was the pre-processing stuff we did.

So we looked at the different modalities and considered what are the artifacts that come

in through the acquisition process and how can we correct for that.

Basically we look into the problem of 3D reconstruction.

So how can I use multiple images acquired by a medical imaging device?

How can I reconstruct three-dimensional information out of that?

Right after Christmas we will also see how 4D reconstruction can be done that includes

dynamic information, just think about the beating heart that has to be reconstructed.

And last but not least we will talk about fusion and image registration.

So that is the story for the winter semester and well, if I ask you what different modalities

do you know, well, just remember the typical ones like x-rays, like CT, like ultrasound,

like MR, like PET, like SPECT and many more.

So there are many, many different modalities used in the medical environment and each modality

has its own advantages and disadvantages.

In terms of preprocessing we have not considered that much, basically we looked at x-ray imaging

and we discussed on the one hand image undistortion and on the other hand we talked about defect

pixel interpolation.

And then we talked about MR inhomogeneities that can be eliminated by various techniques.

And then we started to talk about 3D reconstruction, so if I have x-ray images of excellent quality

can I use x-rays from different directions to do a reconstruction?

And the answer is yes I can.

And how can we do that?

Well, we have seen that there is one important theoretical result, that's the Fourier slice

theorem, that gives us the relationship between the 2D Fourier transform of the slice we want

to reconstruct and the 1D Fourier transform of the detector line.

So we can lift this result also to higher dimensions, the 3D Fourier transform correlates

with a 2D Fourier transform of 2D projections.

So we talked about the Fourier slice theorem and we also looked into the details of the

Fourier slice theorem and it turned out that with the usage of the Fourier slice theorem

we run into a serious problem with different representations of coordinates.

We have seen that polar coordinates are basically the ones we use to collect or to sample the

2D Fourier transform and so to re-transform or to back-transform the 2D Fourier transform

to get the slice we have to do a coordinate transform, a re-gridding to make sure that

we get the slice in Cartesian coordinates.

And if we do that in all the formulas it turned out that the reconstruction by itself is nothing

else but a convolution of the observed signal and then a numerical integration over all

the projection angles that we have.

That's basically the so-called filtered back projection.

And what we have implicitly assumed is we always assume parallel projection.

So we always assume that our detector goes this way and our X-ray beams go this way.

And we all know that if we have an X-ray tube our X-rays and the paths they behave a little

more differently so we have a focus and we have this type of projections.

And so it is a little different.

You cannot apply the Fourier slice theorem or the filtered back projection right away